How to prove an identity by induction

From the inductive step we have $(1+x)^{l-1}=\frac1l\sum_{k=1}^lk\binom lkx^{k-1}$.

Thus$$\begin{align*}(l+1)(1+x)^l&=(1+x)\left[(l+1)(1+x)^{l-1}\right]\\&=(1+x)\left[\left(\frac{l+1}l\right)\sum_{k=1}^lk\binom lkx^{k-1}\right]\\&=\left[\left(\frac{l+1}l\right)\sum_{k=1}^lk\binom lkx^{k-1}\right]+x\left[\left(\frac{l+1}l\right)\sum_{k=1}^lk\binom lkx^{k-1}\right]\end{align*}$$The coefficient of $x^{k},1\le k\le l-1$,$$a_k=\left(\frac{l+1}l\right)\left[(k+1)\binom l{k+1}+k\binom lk\right]=(k+1)\binom{l+1}{k+1}$$which matches with the statement we need to prove. Similarly you can check the coefficients of $x^{0},x^l$.